k-

. 1

k- .-

– –k

k-–

: k-

1.

k

ijx j - i - 1, ij n , 1,i k

i - ( )iF x

0 1 2: ( ) ( ) ( )kH F x F x F x

1]– 1, 2, 3

[4– 5 – –

k- –6 [7, 8, 9].

10

1( )F x 2 ( )F x

2 1 1 2, 2 1su p ( ) ( )n n n nx

D F x F x .

1

48

1 2,n nD

1]:

2 1 1 22 1

, 1 2 2 21 12 1

1max ( ) max ( )n n n r n s

r n s n

r sD F x F x

n n,

2 1 1 22 1

, 1 2 2 11 12 1

1max ( ) max ( )n n n r n s

r n s n

r sD F x F x

n n,

2 1 2 1 2 1, , ,max ,n n n n n nD D D .

0H -

2 11 2

C ,1 2

n nn n

S Dn n

(1)

( )K S [1] .– , 2],

1]

2 1 22 1 22 1

1 11 2 1 2

4 11

( ) 6 ()

n n

i ji j

n nT n r i n s j

n n n n, (2)

ir – 2ix ; js – 1 jx

[3 1( )a t .– 5

1 22

11 2 12

11 2 1 2

( )1

( )

n ni

i

M n n n iA

n n i n n i, (3)

iM – i-

3

0H 2( )a t .

2. k- –

k-– – k-

–11 –

6].

0 1 2: ( ) ( ) ( )kH F x F x F x . ijx j - i -

1, ij n , 1,i k , i -

( )iF x . i -

( )iinF x

1

k

ii

n n ( )nH x k- –

49

22

1

[ ( ) ( )]( )

1 ( ) ( )i

n

k in nkn i n

i B n n

F x H xA n dH x

H x H x,

{ : ( ) 1}n nB x R H x ( )iF x

1 2 nZ Z Z

[6]:2

12

1 1

1 1

( )

k n ij ikn

i ji

nM jnA

n n j n j,

ijM – i - jZ

0H

6]

2

2

( 1)

[ ]

knkn

kn

A kT

D A, (4)

2knA [6]

3 22

( 1)( 2)( 3)knan bn cn d

D An n n

(4 6)( 1) (10 6 )a g k g H ,

2(2 4) 8 (2 14 4) 8 4 6b g k hk g h H h g ,

2(6 2 2) (4 4 6) (2 6) 4c h g k h g k h H h ,

2(2 6) 4d h k hk ,

1

1k

i i

Hn

,1

1

1n

ih

i,

2 1

1 1

1

( )

n n

i j ig

n i j.

4k 1

k 12, 13,14

2 11k-

0 1 0 101 1

2 4 4 42

3 3 33 0 1

( ) 1 1 ( 1)( , )

x x xf x (5)

III 0 1 2 3 4( , , , , )610N

50

4)

1 III 0 1 2 3 4( , , , , )

4 valuep k

4)

k

2 IIIB (3.1575, 2.8730, 18.1238, 15.0000, –1.1600)

3 IIIB (3.5907, 4.5984, 7.8040, 14.1310, –1.5000)

4 IIIB (4.2657, 5.7035, 5.3533, 12.8243, –1.7500)

5 IIIB (6.2992, 6.5558, 5.6833, 13.010, –2.0640)

6 IIIB (6.7446, 7.1047, 5.0450, 12.8562, –2.2000)

7 IIIB (6.7615, 7.4823, 4.0083, 11.800, –2.3150)

8 IIIB (5.8057, 7.8755, 2.9244, 10.900, –2.3100)

9 IIIB (9.0736, 7.4112, 4.1072, 10.800, –2.6310)

10 IIIB (10.2571, 7.9758, 4.1383, 11.186, –2.7988)

11 IIIB (10.6848, 7.5950, 4.2041, 10.734, –2.8400)

N(0.0, 1.0)

3

7, 8, 9– –

2k .

1 2, , ,ii i inx x x

( )iF x , ( 1,i k 1 2 nX X X1

k

ii

n n

ijR j - ijx i -

0X , 1nX ,0 1iR , , 1 1ii nR n .

51

ˆ ( )F t ,

mX , 1,m n ˆ ( ) ( 0.5) /mF X m n [7].

Z 7]:

, ,, ,

1 1

1max ln 1 ln

1

ki m i m

i i m i mm n i m m

F FZ n F F

F F, (6)

ˆ ( )m mF F X ( 0.5)mF m n ,ˆ ( )i m i mF F X

0ij , 1,i k , 1ii jR m

: 1i ij j , ( 0.5)i m i iF j n , , 1i ii j i jR m R ,i m i iF j n .

0H6).

( k 20in

Z k 7]:

, , , ,

1 1

ln 1 ln 1

( 0.5)( 0.5)

n k i m i m i m i mi

m i

F F F FZ n

m n m, (7)

mF ,i mF

0H7).

4 5.

CZ k 7]:

1 1 ,

1ln 1 ln 1

0.5 0.5

inki

Ci j i j

n nZ

n j R. (8)

0H8).

–

valuep .

52

valuep

k 20in

4. k

k -– – 11]. k -

– 1516 k -

– 6

kS ( 1) / 2k k

k -

max ,1max { }i j

i ki j k

S S , (9)

,i jS –

i - j -

0H maxS .

,i jS

[17] – –

maxS

k - ,i jS 9

[17]

2 1

1 2 1 2mod ,

1 2 1 24.6n nn n n n

S Dn n n n

, (10)

53

( )K S .

maxS maxSmS .

maxSmS

67

in

0( )G S H maxSmS

maxSmS , 1000in 7

maxSmS

– 610N ,

maxSmS 3 11k

maxSmS

k

2 ( )K S

3 IIIB (6.3274, 6.6162, 2.8238, 2.4073, 0.4100)

4 IIIB (7.2729, 7.2061, 2.6170, 2.3775, 0.4740)

5 IIIB (7.1318, 7.3365, 2.4813, 2.3353, 0.5630)

6 IIIB (7.0755, 8.0449, 2.3163, 2.3818, 0.6320)

7 IIIB (7.7347, 8.6845, 2.3492, 2.4479, 0.6675)

8 IIIB (7.8162, 8.9073, 2.2688, 2.4161, 0.7120)

9 IIIB (7.8436, 8.8805, 2.1696, 2.3309, 0.7500)

10 IIIB (7.8756, 8.9051, 2.1977, 2.3280, 0.7900)

11 IIIB (7.9122, 9.0411, 2.1173, 2.2860, 0.8200)

III 0 1 2 3 4( , , , , ) - -

(5) 9 ,i jS

10 valuep k

54

k - – ,i jS

maxLRS (9

0H 8.

maxLRS

3 11k 3Sb–

2

31 20 1

2 33 2 3

1( ) exp ln

22 ( )( )

xf x

xx x

3

0 1 2 3 4( , , , , )Sb maxLRS

k valuep .

maxLRS

k

2 1( )a t

3 Sb(3.2854, 1.2036, 3.0000, 0.0215)

4 Sb(2.5801, 1.2167, 2.2367, 0.0356)

5 Sb(3.1719, 1.4134, 3.1500, 0.0320)

6 Sb(2.9979, 1.4768, 2.9850, 0.0380)

7 Sb(3.2030, 1.5526, 3.4050, 0.0450)

8 Sb(3.2671, 1.6302, 3.5522, 0.0470)

9 Sb(3.4548, 1.7127, 3.8800, 0.0490)

10 Sb(3.4887, 1.7729, 3.9680, 0.0510)

11 Sb(3.4627, 1.8168, 3.9680, 0.0544)

k - – ,i jS

maxADS (9 3

0H 9.

max 0( )ADG S H

maxADS 3 11k

- -

III 0 1 2 3 4( , , , , )

valuep k

55

8 maxLRS 9 max

ADS

maxADS

k

2 2( )a t

3 IIIB (4.4325, 2.7425, 12.1134, 8.500, 0.1850)

4 IIIB (5.2036, 3.2160, 10.7792, 10.000, 0.2320)

5 IIIB (5.7527, 3.3017, 9.7365, 10.000, 0.3000)

6 IIIB (5.5739, 3.4939, 7.7710, 10.000, 0.3750)

7 IIIB (6.4892, 3.6656, 8.0529, 10.500, 0.3920)

8 IIIB (6.3877, 3.8143, 7.3602, 10.800, 0.4800)

9 IIIB (6.7910, 3.9858, 7.1280, 11.100, 0.5150)

10 IIIB (6.7533, 4.2779, 6.6457, 11.700, 0.5800)

11 IIIB (7.1745, 4.3469, 6.6161, 11.800, 0.6100)

5.

1H

1 – - 0H 1H )

- 0H

k -

0H

1H – ,

–

0 0.1 1 1 2H –

0 0 -

1 1.1 3H –

56

2

0 0

1 1 1

( ) ( )1( ) exp 1 exp

3 3 3

x xf x

0 0 1 1

0( )G S H 1( )G S H , 2( )G S H

3( )G S H in 5

0.1 4k .

1H , 2H 3H ( k , in n )

n =20 n =50 n = 100 n = 300 n = 500 n = 1000

1H

maxADS 0.112 0.131 0.165 0.302 0.438 0.706

AD 0.112 0.131 0.164 0.301 0.433 0.701

maxLRS 0.113 0.130 0.162 0.293 0.425 0.686

maxSmS 0.111 0.125 0.151 0.261 0.366 0.605

CZ 0.111 0.126 0.155 0.260 0.368 0.595

Z 0.111 0.127 0.153 0.255 0.360 0.579

Z 0.109 0.121 0.141 0.219 0.300 0.502

2H

CZ 0.106 0.122 0.158 0.306 0.468 0.761

Z 0.107 0.124 0.158 0.305 0.463 0.745

Z 0.106 0.120 0.145 0.249 0.367 0.606

AD 0.104 0.110 0.123 0.180 0.254 0.474

maxADS 0.101 0.104 0.111 0.145 0.195 0.381

maxSmS 0.102 0.105 0.108 0.128 0.153 0.221

maxLRS 0.102 0.103 0.105 0.118 0.135 0.197

3H

Z 0.103 0.107 0.116 0.179 0.274 0.566

CZ 0.103 0.107 0.115 0.173 0.257 0.555

Z 0.103 0.107 0.114 0.161 0.222 0.410

AD 0.102 0.106 0.113 0.143 0.179 0.291

maxSmS 0.103 0.104 0.112 0.138 0.166 0.257

maxADS 0.101 0.103 0.107 0.124 0.147 0.229

maxLRS 0.102 0.102 0.105 0.116 0.130 0.183

maxADS – max

LRS maxSmS CZ Z Z .

–

57

CZ Z Z – maxADS max

SmS maxLRS .

Z CZ

–

–

Z CZ Z – maxSmS max

ADS maxLRS .

,

Z , Z , CZ

6.

(4 k -– 2 11k

valuep

(9) maxSmS ( ), max

LRS ( ), maxADS ( ).

1. ––

2. Lehmann E. L. Consistency and unbiasedness of certain nonparametric tests / E. L. Lehmann // Ann. Math. Statist. 1951. Vol. – P. 165–179.

3. Rosenblatt M. Limit theorems associated with variants of the von Mises statistic / M. Rosen-blatt // Ann. Math. Statist. 1952. Vol. 23. – P. 617–623.

4.–

. . . . - . 2007. – . 40-52.5. Pettitt A.N. A two-sample Anderson-Darling rank statistic // Biometrika. 1976. Vol. 63. No.1.

P. 161-168.6. Scholz F.W., Stephens M.A. K-Sample Anderson–Darling Tests // Journal of the American Sta-

tistical Association. 1987. Vol. 82. No. 399. – P. 918-924.7. Zhang J. Powerful goodness-of-fit and multi-sample tests / J. Zhang // PhD Thesis. York Uni-

versity, Toronto. 2001. – 113 p. URL:http://www.collectionscanada.gc.ca/obj/s4/f2/dsk3/ftp05/NQ66371.pdf ( . 28.01.2013).

8. Zhang J. Powerful Two-Sample Tests Based on the Likelihood Ratio / J. Zhang // Technometrics. 2006. V. 48. No. 1. – P.95-103.

9. Zhang J., Wu Y. k-Sample tests based on the likelihood ratio // Computational Statistics & Data Analysis. – 2007. – V. 51. – No. 9. – P. 4682-4691.

58

10.–

3–14.11. Kiefer J. K-Sample Analogues of the Kolmogorov-Smirnov and Cramer-v. Mises Tests // An-

nals of Mathematical Statistics, 1959. Vol. 30. No. 2. – P. 420-447.12.

/

– -31.13. Lemeshko B. Y. Application of Homogeneity Tests: Problems and Solution / B. Y. Lemeshko,

I. V. Veretelnikova, S. B. Lemeshko, A. Y. Novikova // In: Rykov V., Singpurwalla N.,Zubkov A. (eds) Analytical and Computational Methods in Probability Theory. ACMPT 2017. Lecture Notes in Computer Science. : monograph. - Cham : Springer, 2017. - 10684. - P. 461-475.

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15. Conover W. J. Several k-sample Kolmogorov-Smirnov tests // The Annals of Mathematical Statistics. 1965. Vol. 36, No. 3. – P.1019-1026.

16. Conover W. J. Practical Nonparametric Statistics / W. J. Conover. – 3d ed. – Wiley, 1999. – 584p.

17.–– –14.

(630073 ), (383) 346-06-00, e-mail: Lemeshko@ami.nstu.ru, http://www.ami.nstu.ru/~headrd/

(383) 346-06-00, e-mail: ira-veterok@mail.ru.

About application and power of k-samples homogeneity tests

B. Yu. Lemeshko, I. V. Veretelnikova

Novosibirsk State Technical University

Properties of k-samples tests for homogeneity of distribution laws have been investigated. Tests have been proposed in which the statistics are being the maximum of the two-samples statistics of the Smirnov, the Lehmann-Rosenblatt and the Anderson-Darling tests of pairs of the k-samples. Models of limiting statistics distributions for the proposed tests, as well as for the An-derson-Darling k-sample test, have been made. Comparative analysis of the power of the tests has been done.

Keywords: k-samples tests, homogeneity test, power of test, statistical simulating

59